# American Institute of Mathematical Sciences

December  2020, 7(2): 369-399. doi: 10.3934/jcd.2020015

## A tale of two vortices: How numerical ergodic theory and transfer operators reveal fundamental changes to coherent structures in non-autonomous dynamical systems

 School of Mathematics and Physics, The University of Queensland, St Lucia, QLD 4072, Australia

* Corresponding author: uqcblach@uq.edu.au

Received  October 2019 Published  July 2020

Fund Project: This work has been partially supported by an Australian Research Council Discovery Early Career Researcher Award (DE160100147) and by an Australian Government Research Training Program Stipend Scholarship (CB)

Coherent structures are spatially varying regions which disperse minimally over time and organise motion in non-autonomous systems. This work develops and implements algorithms providing multilayered descriptions of time-dependent systems which are not only useful for locating coherent structures, but also for detecting time windows within which these structures undergo fundamental structural changes, such as merging and splitting events. These algorithms rely on singular value decompositions associated to Ulam type discretisations of transfer operators induced by dynamical systems, and build on recent developments in multiplicative ergodic theory. Furthermore, they allow us to investigate various connections between the evolution of relevant singular value decompositions and dynamical features of the system. The approach is tested on models of periodically and quasi-periodically driven systems, as well as on a geophysical dataset corresponding to the splitting of the Southern Polar Vortex.

Citation: Chantelle Blachut, Cecilia González-Tokman. A tale of two vortices: How numerical ergodic theory and transfer operators reveal fundamental changes to coherent structures in non-autonomous dynamical systems. Journal of Computational Dynamics, 2020, 7 (2) : 369-399. doi: 10.3934/jcd.2020015
##### References:

show all references

##### References:
Figures 1a and 1b show almost invariant structures as described by the (evolved) subdominant eigenvector of an Ulam matrix approximation to the transfer operator in the periodically driven double gyre flow, with parameters as in [40]. Figures 1c and 1d show finite-time coherent structures as described by the (evolved) subdominant initial time singular vector of a composition of $10$ Ulam matrices describing the evolution of the transitory double gyre flow introduced by [31]. See [17] for a thorough discussion of both models
Evolution in non-autonomous dynamical systems: driving system (above the arrow), particle evolution (2nd row), transfer operators (3rd row) and Ulam's method (bottom row)
Selected vector field instances for the periodically forced double well potential
An illustration of the behaviour of $\alpha(t)$ and $\tilde{\alpha}(t)$ over $5$ periods
Tracking modes over rolling windows for the periodically forced double well potential
Tracking modes for time windows of length $n = 50$, evolved using Algorithm 4
Crossing introduced by shifting from $n = 54$ to $n = 51$ for the periodically forced double well potential
Equivariance mismatch for the periodically forced double well potential when $n = 50$
Leading $6$ of $\mathcal{N} = 6$ modes for the periodically forced double well potential when $n = 50$
Leading $4$ of $\mathcal{N} = 4$ modes for the periodically forced double well potential when $n = 100$
Mean equivariance mismatch, as per Algorithm 5, for the leading $4$ of $\mathcal{N}$ modes using the two pairing methods given by Algorithms 2 ($\bar{\varsigma}_{S}$) and 3 ($\bar{\varsigma}_{U}$) for $n = 50$
Leading $4$ from a total $\mathcal{N} = 5$ tracked modes for $n = 50$ using Algorithms 3 (top) and 5 (bottom)
Leading $4$ from a total $\mathcal{N} = 7$ tracked modes for $n = 50$ using Algorithms 3 (top) and 5 (bottom)
Consecutive windows corresponding to reasonable equivariance for $S_{U}^{(4)}$ of Figure 12
Initial time singular vectors corresponding to rolling windows initialised at the various $t_{0}$ indicated by colour coded bars and column headings. These are paired according to the paths illustrated in Figure 12
Evolved $u^{(50)}_{75,4}$ (top) and $u^{(50)}_{274,4}$ (bottom) of mode $S_{U}^{(4)}$ in Figure 15, evolved as per Algorithm 4
Southern hemisphere wind speed (easterly and northerly) on the $850$ K isentropic surface
Mean equivariance mismatch, as per Algorithm 5, for the leading $3$ of $\mathcal{N}$ modes using the two pairing methods given in Algorithms 2 and 3 with $n = 56$ and $t_0 \in[0000 \: 1 \: \text{August}, 1800 \: 30 \: \text{September}]$. Here the Ulam matrices, describing transitions for the area south of $30^{\circ}$S, are of dimension $m \times m$ for $m = 2^{14}$
Leading $3$ of $\mathcal{N} = 3$ tracked paths of singular values of rolling windows paired using Algorithm 2 for $n = 56$
Leading singular vectors, for various $t_{0}$, of matrix compositions associated with Figure 19b where time windows are of length $n = 56$. The area illustrated is south of $50^{\circ}$S and the time given in the label is the relevant $t_{0}$ for that window
Evolved leading mode associated with Figure 19a for a time window centred on the peak at $1800$ on $23$ Sep. This is illustrated on the area south of $15^{\circ}$S
Evolved subdominant mode associated with Figure 19b for a time window centred on the peak at $0600$ on $24$ Sep. This is illustrated on the area south of $15^{\circ}$S
Evolved leading singular vectors for time windows centred at $0000$ on $24$ Sep. for $m = 12,800$ initially seeded bins whose centres are south of $20^{\circ}$S. This is illustrated on the full southern hemisphere
Evolved subdominant mode normalised as in [19] for time windows centred at $0000$ on $24$ Sep. for $m = 12,800$ initially seeded bins whose centres are south of $20^{\circ}$S. This is illustrated on the full southern hemisphere
 [1] Michael Dellnitz, Christian Horenkamp. The efficient approximation of coherent pairs in non-autonomous dynamical systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3029-3042. doi: 10.3934/dcds.2012.32.3029 [2] Mikhail B. Sevryuk. Invariant tori in quasi-periodic non-autonomous dynamical systems via Herman's method. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 569-595. doi: 10.3934/dcds.2007.18.569 [3] Alexandre N. Carvalho, José A. Langa, James C. Robinson. Non-autonomous dynamical systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 703-747. doi: 10.3934/dcdsb.2015.20.703 [4] Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087 [5] Grzegorz Łukaszewicz, James C. Robinson. Invariant measures for non-autonomous dissipative dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4211-4222. doi: 10.3934/dcds.2014.34.4211 [6] Michael Zgurovsky, Mark Gluzman, Nataliia Gorban, Pavlo Kasyanov, Liliia Paliichuk, Olha Khomenko. Uniform global attractors for non-autonomous dissipative dynamical systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 2053-2065. doi: 10.3934/dcdsb.2017120 [7] Tomás Caraballo, David Cheban. On the structure of the global attractor for non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2012, 11 (2) : 809-828. doi: 10.3934/cpaa.2012.11.809 [8] Rua Murray. Ulam's method for some non-uniformly expanding maps. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 1007-1018. doi: 10.3934/dcds.2010.26.1007 [9] David Cheban, Cristiana Mammana. Continuous dependence of attractors on parameters of non-autonomous dynamical systems and infinite iterated function systems. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 499-515. doi: 10.3934/dcds.2007.18.499 [10] Emma D'Aniello, Saber Elaydi. The structure of $\omega$-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195 [11] Alexandre N. Carvalho, José A. Langa, James C. Robinson. Forwards dynamics of non-autonomous dynamical systems: Driving semigroups without backwards uniqueness and structure of the attractor. Communications on Pure & Applied Analysis, 2020, 19 (4) : 1997-2013. doi: 10.3934/cpaa.2020088 [12] Noriaki Yamazaki. Global attractors for non-autonomous multivalued dynamical systems associated with double obstacle problems. Conference Publications, 2003, 2003 (Special) : 935-944. doi: 10.3934/proc.2003.2003.935 [13] Tomás Caraballo, David Cheban. On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2013, 12 (1) : 281-302. doi: 10.3934/cpaa.2013.12.281 [14] Bixiang Wang. Multivalued non-autonomous random dynamical systems for wave equations without uniqueness. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 2011-2051. doi: 10.3934/dcdsb.2017119 [15] Carmen Núñez, Rafael Obaya. A non-autonomous bifurcation theory for deterministic scalar differential equations. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 701-730. doi: 10.3934/dcdsb.2008.9.701 [16] Pablo G. Barrientos, Abbas Fakhari. Ergodicity of non-autonomous discrete systems with non-uniform expansion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1361-1382. doi: 10.3934/dcdsb.2019231 [17] Christopher Bose, Rua Murray. The exact rate of approximation in Ulam's method. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 219-235. doi: 10.3934/dcds.2001.7.219 [18] Flank D. M. Bezerra, Vera L. Carbone, Marcelo J. D. Nascimento, Karina Schiabel. Pullback attractors for a class of non-autonomous thermoelastic plate systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3553-3571. doi: 10.3934/dcdsb.2017214 [19] Mahesh G. Nerurkar. Spectral and stability questions concerning evolution of non-autonomous linear systems. Conference Publications, 2001, 2001 (Special) : 270-275. doi: 10.3934/proc.2001.2001.270 [20] Ming-Chia Li, Ming-Jiea Lyu. Topological conjugacy for Lipschitz perturbations of non-autonomous systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5011-5024. doi: 10.3934/dcds.2016017

Impact Factor: